Download Impact of Tandem Repeats on the Scaling of Nucleotide Sequences

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Impact of Tandem Repeats on the Scaling of Nucleotide Sequences
Radhakrishnan Nagarajan*
Center on Aging, University of Arkansas for Medical Sciences
Meenakshi Upreti
Department of Biochemistry and Molecular Biology, University of Arkansas for Medical Sciences
*To whom correspondence should be addressed
Radhakrishnan Nagarajan
Center on Aging, University of Arkansas for Medical Sciences
629 Jack Stephens Drive, Room: 3105
Little Rock, Arkansas 72205
Phone: (501) 526 7461
Email: [email protected]
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ABSTRACT
Techniques such as detrended fluctuation analysis (DFA) and its extensions have been widely
used to determine the nature of scaling in nucleotide sequences. In this brief communication we
show that tandem repeats which are ubiquitous in nucleotide sequences can prevent reliable
estimation of possible long-range correlations. Therefore, it is important to investigate the
presence of tandem repeats prior to scaling exponent estimation.
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1. Introduction
Understanding patterns in eukaryotic DNA sequences is an area of active research. This is more
so with the rapid completion of eukaryotic genomes. DNA sequences are composed of four
nucleotides (A, G, C and T), with (A, G) representing the purines and (C, T) the pyrimidines.
Repetitive nucleotide patterns form a prominent part of eukaryotic genomes and manifest
themselves as tandem repeats. Such repeats are usually approximate repeats adjacent to each
other and include microsatellites, minisatellites, CpG islands, and telomeric repeats [Dogett et al.,
1992; Toth et al., 2000]. Classical techniques such a Fourier analysis have been used to identify
short-term correlated patterns in DNA sequences [Silverman & Linsker, 1986; Tavare &
Giddings, 1989; Coward, 1997]. Such correlations are of finite memory or Markovian in nature,
and can be broadly classified into periodic and quasi-periodic repeats. However, recent studies
have provided compelling evidence of non-Markovian characteristics in the form of long-range
correlations (LRC) in DNA sequences [Peng et al., 1992; Li and Kaneko, 1992]. Such
correlations persist over large nucleotide distances, also known as time scales. This often results
in power-law spectral signatures of the form S ( f ) ~ f
−β
, where frequency is represented as the
reciprocal of basepairs (bp) , i.e. (1/bp). Several algorithms have been proposed in the past to
determine the scaling exponent of a given sequence [Feder, 1988; Li & Kaneko, 1992; Peng et al.,
1992]. Detrended fluctuation analysis DFA and its extensions multifractal DFA (MF-DFA) [Peng
et al., 1992; Kantelhardt et al., 2002] have been successfully used in the past to estimate the
scaling exponents in sequences obtained from diverse settings (Hu et al., 2001 and references
there in). DNA sequences are composed mainly of coding regions (exons) which code for specific
proteins and non-coding regions (introns) which are interspersed between exons. Earlier studies
provided overwhelming evidence of LRCs in the introns (α > 0.5) and absence of correlations in
exons (α = 0.5) [Li & Kaneko, 1992; Peng et al., 1992]. Subsequently, several independent
studies have used DFA to arrive at similar results.
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In the present study, we show that estimation of the scaling exponent can be significantly
affected by the presence of tandem repeats which are ubiquitous in DNA sequences. While
scaling in introns have been associated with the presence of long-range correlations (α > 0.5)
drawing such conclusion can be non-trivial in the presence of tandem repeats. This is attributed to
characteristic distortion introduced by the tandem repeats in the log-log plot of the fluctuation
function versus time scale, rendering the estimation of the scaling exponent unreliable . Therefore,
it is important to examine a given nucleotide sequence for tandem repeats prior to scaling
exponent estimation. The present study also raises new concerns on interpreting results obtained
on genes and chromosomes, which are riddled with such repeats.
Recent reports have investigated the impact of stationary uncorrelated trends superimposed on
power-law noise [Hu et al., 2001; Nagarajan & Kavasseri, 2005; Nagarajan & Kavasseri, 2005].
However, there are subtle differences between stationary uncorrelated trends and tandem repeats.
Tandem repeats occur as patches hence non-stationary in the sequence space. Such repeats can
also be an inherent part of the dynamics, hence may not fall under the class of uncorrelated
trends. In a recent study, GC rich tandem repeats have been found to exhibit oscillations
characteristic of deterministic nonlinear dynamical systems [Nicolay et al., 2004]. Therefore,
extending earlier results [Hu et al., 2001; Nagarajan & Kavasseri, 2005; Nagarajan & Kavasseri,
2005] on the impact of uncorrelated trends on power-law noise to tandem repeats is not
immediate.
2. Methods
The given nucleotide sequence of alphabet size four (A, G, C, T) is converted into an indicator
sequence with alphabet size two by the mapping (A, G) → -1 and (C, T) → 1 [Peng et al.,
1992]. A brief description of the DFA (MF-DFA) procedure is enclosed below for completeness.
A detailed treatment can be found elsewhere [Peng et al., 1992; Kantelhardt et al., 2002].
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Description of DFA and MF-DFA:
a. Determine the integrated series from the given indicator sequence {x k }, k = 1...n as
i
y i = ∑ ( x k − µ ), i = 1...n where µ represents the average value of {x k }, k = 1...n
k =1
n
b. The data is divided into n s non-overlapping bins of size s, where n s =   ; 
s
represents the greatest integer not larger than

n
. The local trend yυ (i ) is calculated is
s
calculated in each of the bins ( υ = 1…n s ) by a polynomial regression and the variance is
determined, given by
F (υ , s ) =
2
1 s
{y[(υ − 1) s + i] − yυ (i)}2
∑
s i =1
Recent studies have suggested higher order polynomia l regression in order to minimize
the effect of local trends [Ashkenazy et al., 2001; Kantelhardt et al., 2001].
c. This procedure is repeated from either end of the data in order to accommodate all the
samples. Thus the effective length of the data is 2n s .
d. The fluctuation F 2 (υ , s) is averaged over to obtain the q th order fluctuation function
given by
 1
Fq ( s ) = 
 2n s
2ns
∑F
υ =1
2
(υ , s )
q/ 2



1/ q
where q can take any real value ( q ≠ 0) . The scaling behavior is determined by
analyzing the log-log plots of Fq (s ) versus s.
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The DFA code for estimating the scaling exponent is publicly available and can be obtained from
(http://www.physionet.org/physiotools/dfa/).
The scaling exponent(s) of
{x k }, k = 1...n is
determined from the log-log plot of fluctuation
function Fq (s) versus time scale (s) with varying moments (q, q ∈ R and q ≠ 0). Monofractal
sequences exhibit a single exponent and their fluctuation function varies linearly with time scale
in the log scale . Therefore, determining the variation of Fq (s) versus (s) for a single moment (q =
2) is sufficient to capture the ir scaling behavior. This has to be contrasted with multifractals
whose fluctuation function varies as a function of moments q. It should be noted that
investigating the qualitative dynamics of the fluctuation function versus time scale with varying
moments can be useful in discerning monofractal from multifractal dynamics in the presence of
trends [Nagarajan and Kavasseri, 2005].
To establish the effect of tandem repeats on scaling exponent estimate, we compare the scaling
behavior of introns with tandem repeats such as (CpG islands) [Gardiner-Garden & Frommen,
1987; Aissani & Bernardi, 1991; Larsen et al., 1992] to those without tandem repeats. CpG
islands are GC rich regions which are resistant to methylation, hence directly associated with the
activity of genes. Such regions are ubiquitous in house keeping and tissue specific genes, hence
their choice is valid within the present context.
We consider introns from (Human blood
coagulation factor VII (HUMCFVII, Genbank ID: J02933, 12850 basepairs) [O’Hara et al., 1987]
and Homo Sapiens limb-girdle muscular dystrophy type 2b gene (LGMD2B, Genbank ID:
AJ007973, 36133 basepairs) [Bashir et al., 1998]. The choice of HUMCFVII is encouraged by Li
and Kaneko’s seminal report on the existence of partial power-law decay and LRCs in this gene.
HUMCFVII has nine exons and eight introns located at (522:585; 1654:1719; 4294:4454;
6383:6407; 6478:6591; 8307:8447; 9419:9528; 10124:10247; 11064:11659) and (586:1653;
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1720:4293; 4455:6382; 6408:6477; 6592:8306; 8448:9418; 9529:10123; 10248:11063)
respectively. Investigating HUMCFVII for possible GC rich domains (CpG plot, EMBOSS,
http://www.ebi.ac.uk/emboss/cpgplot/ with parameters Observed/Expected ratio = 0.6; length >
200; Percent C + Percent G = 50%) revealed three distinct regions : (2075:2677), (2996:4726),
(11050:11313) with lengths 603, 1731 and 264 respectively. Two of the GC rich domains
(2075:2677 and 2996:4726) had a significant overlap with the intron located at (1720:4293). In
the subsequent discussion we shall refer to this GC rich intron as (IntronREP , N = 2574
nucleotides). LGMD2B gene consists of three exons and three introns located at (19078:19176;
31270:31398; 33986:34141; 35668:35759) and (19177:31269; 31399:33985; 34142:35667)
respectively. Investigating the intron (19177:31269) using CpG plot with the same parameters as
above failed to reveal any GC rich domains. Therefore, we use this intron as the reference
sequence. In order reject the claim that the observed distortion (crossover) in scaling is an
outcome of finite sample size effects, we chose the region (19177+1720:19177+4293) which is of
the same length as IntronREP (N = 2574). Since this intron does not contain any GC rich regions
we shall refer to it as (IntronNOREP ) in the subsequent discussion.
3. Results
Spectral signatures of IntronNOREP and IntronREP exhibited characteristic low-frequency power-law
decay, Fig. 1, conforming to earlier observations of LRC in introns [Li & Kaneko., 1992; Peng et
al., 1992]. In addition, IntronREP exhibited prominent peaks which were absent in IntronNOREP , Fig.
1. These confirm our earlier observation of patchy GC rich domains in IntronREP unlike
IntronNOREP . Investigating the log-log plot of the fluctuation function F2 (s) versus sequence length
(s) revealed characteristic crossover for IntronREP indicating possible existence of more than one
scaling exponent, Fig. 2a. The nonlinear nature of the log-log plot also prevented reliable
estimation of the scaling exponent. This has to be contrasted with that of IntronNOREP whose loglog plot was linear with a scaling exponent (α ~ 0.6). In order to reject the claim that the
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crossover in the case of IntronREP is due to local polynomial trends in the integrated series,
{y k }, k = 1...n , Sec. 2, we invest investigated the log-log plot of F2(s) versus (s) of IntronNOREP
and IntronREP with various orders of polynomial detrending (d = 1, 2, 3 and 4), Fig. 2. The log-log
plots with (d = 1, 2, 3 and 4) failed to reveal any characteristic change in the scaling behavior
indicating that the observed crossover is not an outcome of local polynomial trends in the
integrated series.
The results obtained using DFA on IntronNOREP and IntronREP were also confirmed with more
classical tools such as rescaled-range (R/S) analysis [Feder, 1988], Fig. 3. While the scaling of
IntronNOREP was linear with exponent (α ~ 0.6) those of IntronREP was nonlinear with a marked
crossover, Fig. 2.
4. Discussion
In this brief note, we showed that tandem repeats can have a significant impact on reliable
estimation of possible long-range correlations in DNA sequences. Such repeats can introduce
crossovers and nonlinearity in scaling of nucleotide sequences even at the level of introns.
Several reports have been published in the past on the scaling behavior of introns, exons, genes
and chromosomes. However, such sequences are riddled with various types of tandem repeats.
The present study encourages investigation of tandem repeats as an important preliminary step
prior to scaling exponent estimation.
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Figure 1 Power spectral density of IntronREP (dotted line) and IntronNOREP (solid line).
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Figure 2 Log-log plot (log2 scale) of the fluctuation function F2 (s) versus sequence length (s) of
IntronREP (a) and IntronNOREP (b) with various orders of polynomial detrending (d = 1, 2, 3 and 4,
from top to bottom, in that order). The arrow in (a) indicates crossover in the scaling behavior.
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Figure 3 Log-log plots (log2 ) of (R/S) versus sequence length (s) of IntronNOREP (a) and IntronREP
(b) obtained using rescaled-range analysis. The arrow in (a) indicates crossover in the scaling
behavior.
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